[email protected] wrote:Of the guests at a charity fundraiser, 180 own both a house in the Hamptons and a house in Palm Beach. If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach, what is the ratio of the number of people who own a house in Palm Beach but not in the Hamptons to the number of people who own a house in the Hamptons but not in Palm Beach?

(1) One-half of the guests own a house in Palm Beach.

(2) Two-thirds of the guests own a house in the Hamptons.

My answer:
X guests = hous in hampton + house in palm beach - hampton & palm
X guests = (2/3)x + (1/3)x - 180 -> sufficient

Pls tell me where I am wrong

Hi, there. I'm happy to help!

I agree with your basic set up. I am going to use N for the total number of people at the fundraiser.

Statement #1: (1/2)N own in Palm Beach (of course, dahling!)

Statement #2: (2/3)N own in the Hamptoms (who doesn't!)

Both of those are clearly insufficient on their own. Now, together, we can say

(# who own in PB or H) = (1/2)N + (2/3)N - 180

The problem is --- and this is your mistake --- the set "people who own in either the Hamptons or in Palm Beach" is not coextensive with the entire set, and they even say so in the question: "not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach."

Thus,

N = (1/2)N + (2/3)N - 180 + X

where X is the number of those poor unfortunate lowlifes who are so gauche, so unrefined, that they actually don't own a house in either the Hamptons or in Palm Beach. Heavens to Murgatroyd! Dahling, can you even believe there are people like that in the world? OK, sarcasm aside, we have no idea of the count of X, but from the stem, we know it's greater than zero, so that prevents us from solving for N.

Does that make sense? Let me know if you have any more questions.

Mike